On 5/21/2012 6:26 PM, Russell Standish wrote:

On Mon, May 21, 2012 at 07:42:01AM -0400, Stephen P. King wrote:On 5/21/2012 12:33 AM, Russell Standish wrote:On Sun, May 20, 2012 at 12:06:05PM -0700, meekerdb wrote:On 5/20/2012 9:27 AM, Stephen P. King wrote:4) What is the cardinality of "all computations"?Aleph1.## Advertising

Actually, it is aleph_0. The set of all computations is countable. OTOH, the set of all experiences (under COMP) is uncountable (2^\aleph_0 in fact), which only equals \aleph_1 if the continuity hypothesis holds.Hi Russell, Interesting. Do you have any thoughts on what would follow from not holding the continuity (Cantor's continuum?) hypothesis?No - its not my field. My understanding is that the CH has bugger all impact on quotidian mathematics - the stuff physicists use, basically. But it has a profound effect on the properties of transfinite sets. And nobody can decide whether CH should be true or false (both possibilities produce consistent results).

Hi Russell,

`I once thought that consistency, in mathematics, was the indication`

`of existence but situations like this make that idea a point of`

`contention... CH and AoC <http://en.wikipedia.org/wiki/Axiom_of_choice>`

`are two axioms associated with ZF set theory that have lead some people`

`(including me) to consider a wider interpretation of mathematics. What`

`if all possible consistent mathematical theories must somehow exist?`

Its one reason why Bruno would like to restrict ontology to machines, or at most integers - echoing Kronecker's quotable "God made the integers, all else is the work of man".

`I understand that, but this choice to restrict makes Bruno's`

`Idealism even more perplexing to me; how is it that the Integers are`

`given such special status, especially when we cast aside all possibility`

`(within our ontology) of the "reality" of the physical world? Without`

`the physical world to act as a "selection" mechanism for what is "Real",`

`why the bias for integers? This has been a question that I have tried to`

`get answered to no avail.`

This is the origin of Bruno's claim that COMP entails that physics is not computable, a corrolory of which is that Digital Physics is refuted (since DP=>COMP).Does the symbol "=>" mean "implies"? I get confused ...Yes, that is the usual meaning. It can also be written (DP or not COMP).

"=>" = "or not"

`I am still trying to comprehent that equivalence! BTW, I was`

`reading a related Wiki article`

`<http://en.wikipedia.org/wiki/Transposition_%28logic%29> and found the`

`sentence "the truth of "A implies B" the truth of "Not-B implies`

`not-A"". That looks familiar... Didn't I write something like that to`

`Quentin and was rebuffed... I wrote it incorrectly it appears...`

Of course in Fortran, it means something entirely different: it renames a type, much like the typedef statement of C. Sorry, that was a digression.

`That's OK. ;-) I suppose that it is a blessing to be able to "think`

`in code". ;-)`

-- Onward! Stephen "Nature, to be commanded, must be obeyed." ~ Francis Bacon -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.